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Cake
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Preheat the oven to 350 degrees. Spray a 13- by 9-inch baking dish with cooking spray and set aside.
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Combine the cake ingredients in a large mixing bowl and beat with an electric mixer on medium for about 2 minutes, until batter is smooth. Scrape down the sides and bottom of the bowl and beat a few seconds more.
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Pour the batter into the prepared baking dish and bake 30 to 35 minutes, until a cake tester inserted into the center of the cake comes out clean.
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Cool completely on a wire rack before frosting.
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Frosting
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Combine the peanut butter, butter, and milk, in a medium saucepan set over medium heat. Heat mixture, stirring often, until the butter and peanut butter are melted and the mixture is smooth.
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Remove from the heat and add the powdered sugar, vanilla, and salt. Stir vigorously until smooth; it should be fairly thick but still pourable. (The frosting will set up quite a bit in the fridge.)
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Pour over the cooled cake, smooth the top, and sprinkle with chopped peanuts (if desired).
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Refrigerate to set the frosting, 20 to 30 minutes, then slice and serve.
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Store cake in the fridge, covered, for up to 2 days or in the freezer, tightly wrapped, for up to 2 months.